Issue 78

F. Veschi et alii, Fracture and Structural Integrity, 78 (2026) 44-73; DOI: 10.3221/IGF-ESIS.78.04

of peak values and the subsequent cumulative damage evaluation [2]. Due to the non-Gaussianity of the process, the use of standard spectral methods, such as Dirlik's or narrow-band approximation, will lead to inaccurate life estimations [8].

Figure 3: Distribution of instantaneous values of a single sinusoid SoR signal with varying / σ x a ratio.

A VAILABLE APPROACHES FOR S O R FATIGUE DAMAGE EVALUATION

T

o overcome the limitations of classical spectral methods imposed by the non-Gaussian nature of SoR signals, the literature proposes two primary frameworks: the theoretical derivation of the cycle PDF and the practical derivation of an equivalent PSD. Cycle PDF equivalence: Rice's method Originally pioneered by S.O. Rice [10], this theoretical approach focuses on the rigorous derivation of the PDF of the stress cycle amplitudes. It treats Sine-on-Random excitation as a distinct stochastic process, explicitly modelling the interaction between the deterministic harmonic components and the background Gaussian noise. The fundamental objective of this approach is to estimate the PDF of the rainflow cycle amplitudes directly in the frequency domain based exclusively on statistical properties. Rice's approach avoids the forced Gaussian assumption, thereby preserving the inherent non-Gaussian characteristics of the SoR signal. These characteristics include the unique peak distribution and the increased probability of high-amplitude cycles. For a NB random process overlaid with h N harmonics, Rice's PDF for peak values p s is expressed as follows:

2 2 , 2 s r x

 





  p











N





xJ s x e 

J a x dx 

p s

s

(21)

h

s

p

p

S

0



i

1 0

0

p

i

where 2

( 1,2, , h i N   ) are the sine amplitudes and 0 J is the

, σ s r is the variance of the zero-mean random component, i S a

Bessel function of order zero. The only case in which Eqn. (21) offers a closed-form solution is the single-tone scenario: a single sinusoidal signal ( 1 h N  ) of amplitude S a superimposed on a random PSD. The resulting PDF takes the following shape:

2 2 2 S a 

s

p

   

   

s

a s



  p

2 p

S p

s r I 

, 2



p s

e

(22)

s

0

2





p

, s r

, s r

where 0 I represents the modified Bessel function of order zero. Once the PDF is known, the evaluation of fatigue damage can proceed by means of Miner's integral linear damage evaluation formula (Eqn. (10)). If the random component of the SoR process is NB, the SoR peak values PDF   p s p p s represents a

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